A barometric damper is a device that is installed on flue pipe between the exhaust port on your heating appliance and the thimble on the chimney. The function of this device is to help regulate the draft. These are typically installed on any heating appliance with a flue and a chimney with the exception of wood. A barometric damper can be used on any type of coal heating appliance but some prefer a manual damper instead for hand fired coal stoves.
The barometric damper has a swing gate on it, this gate will open when there is excessive draft. It's calibrated by a weight on the face of the gate and it needs to be installed level. When the gate is open this allows room air can be pulled into the flue and chimney. This increases operating efficiency because it prevents air from being pulled through your heating appliance sending a lot of heat out the chimney. The loss of heated room air is far less than heated air from the heating appliance.
For coal stoves, boilers and furnaces this has an additional role. When using coal the amount of fuel available is not the primary regulator for more or less heat production. The typical operation of a hand fired coal stove for example is filling it completely once every 12 hours. The burn rate and the amount of heat produced is regulated by the amount of air. Excessive draft can cause problems with regulating your coal heating appliance properly.
Special word of caution if you have used wood in the same chimney you are going to now use a coal appliance with a barometric damper. Coal itself presents no hazard for a chimney fire but if you have used wood in the past the chimney should be cleaned before using coal. Coal will dry creosote out and it will eventually fall off potentially blocking the flue pipe. This also presents an additional hazard with a barometric damper, if the creosote is to catch fire the barometric damper provides the perfect source of air. This also why they are not used on wood stoves.